Copying and moving works in the Organizer (“Organize” >> “Organizer”). The Organizer is also accessible through the “Organize” option in the various "Tools" menus like Featured Albums, Category, or "Organize Photos" in the "Tools" for photos. Its workspace and all functions are described in the Organizer - general article.

Which targets are valid ?!  Valid targets are also recognizable by the green tick mark  next to the dragged items (as opposed to the red cross mark  for invalid targets) :
a) My albums (homepage) for any category/album already not there;
b) My albums (homepage) for any number of categories/albums selected from the contents panel (right) and not already present on your homepage;
c) a Category for a different category or an album;
d) a Category for any number of categories/albums selected on the right;
e) an Album for any number of photos selected on the right;
g) Featured albums for an album from the navigation panel (left);
f) Featured albums for any number of albums from the contents panel (right);
g) Featured photos for any number of photos from the contents panel (right).

That is nice but how do I move an album from its category to a different category ?  There are two ways: 1. you expand the appropriate category in the navigation panel, catch the album with your mouse (click on it and hold it using the left mouse/touchpad button), drag it into the destination category and once you see a green tick , you can release it. The operation is illustrated with an album named "2013" being moved to the "seeking" category (▼).
The second way is similar to the way photos are copied/moved. Once you see the preferred album) in your contents panel, you drag it to the destination category on the left (▼). After both methods, you have to wait for the operation to successfully finish. That should not take longer than few moments.

Copying photos. If you want to copy photos from one album to another, you have to open the appropriate album e.g. using the navigation panel. Once the photos are displayed in the contents panel, you can select the photos to be copied (▼). You can use  or select the first from the photos and while holding SHIFT select the last of the photos of your preferred selection. While holding CTRL, you may add any further photos.

Then, while making sure your selection holds (the number of item selected is displayed in the upper row - the information panel - of the Organizer), you drag them into the appropriate album in the navigation panel. Once you release them, you will be asked to choose between moving the photos and copying them (►). Depending on the number of photos being copied, the operation may take more than minutes.
The Organizer also allows ordering/arranging of categories/albums/photos, and other options including the management of the Featured albums and Featured photos. Furthermore, it allows multiple photo/video editing, downloading of the uploaded originals, and applying watermarks.

Please note :  the Organizer does not allow you to drag any item from the navigation panel to the contents panel nor from the contents panel into a different item in the contents panel. It is also impossible to move categories and albums at once.